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Syllabus: Sets, Relations, Functions, Partial orders, Lattices, Monoids, Groups.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} &\textbf{2024-2} &\textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&1&0& 1&0&1&0&0.83&1
\\\hline\textbf{2 Marks Count} &1&1&2& 0 &2&1&0&1.16&2
\\\hline\textbf{Total Marks} & 3&3&4&1&4&3&\bf{1}&\bf{3}&\bf{4}\\\hline
\end{array}}}$$

Recent questions in Set Theory & Algebra

#261
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Give an example of two uncountable sets $A$ and $B$ such that $A − B$ isfinite.countably infinite.uncountable.
#262
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Suppose that a countably infinite number of buses, each containing a countably infinite number of guests, arrive at Hilbert's fully occupied Grand Hotel. ... all the arriving guests can be accommodated without evicting any current guest.
#263
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Show that a countably infinite number of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
#264
214
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Suppose that Hilbert's Grand Hotel is fully occupied on the day the hotel expands to a second building which also contains a countably infinite number of rooms. Show ... be spread out to fill every room of the two buildings of the hotel.
#265
219
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Suppose that Hilbert’s Grand Hotel is fully occupied, but the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain in the hotel.
#266
215
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Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.
#267
388
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Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive ... numbers with decimal representations of all $1s\: \text{or}\: 9s$
#268
391
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Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set ... numbers containing only a finite number of $1s$ in their decimal representation
#269
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Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of ... $A = \{2, 3\}$the integers that are multiples of $10$
#270
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Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the ... less than $1,000,000,000$the integers that are multiples of $7$
#271
227
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Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$\displaystyle{}\prod_{i=0}^{4} j\:!$
#272
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Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$\displaystyle{}\sum_{i=0}^{4} j\:!$
#273
188
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Recall that the value of the factorial function at a positive integer $n,$ denoted by $n!,$ is the product of the positive integers from $1$ to $n,$ inclusive. Also, we specify that $0! = 1.$Express $n!$ using product notation.
#274
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What are the values of the following products?$\displaystyle{}\prod_{i=0}^{10} i$\displaystyle{}\prod_{i=5}^{8} i$\displaystyle{}\prod_{i=1}^{100} (-1)^{i}$\displaystyle{}\prod_{i=1}^{10} 2$
#275
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Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.There is also a special notation for products. The product of ... $j = m\: \text{to}\: j = n\: \text{of}\: a_{j} .$
#276
199
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Find a formula for $\displaystyle{}\sum_{k=0}^{m}\left \lfloor \sqrt{k} \right \rfloor ,$ when $m$ is a positive integer.
#277
160
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Find $\displaystyle{}\sum_{k=99}^{200}k^{3}. \text{(Use Table 2.)}$
#278
141
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Find $\displaystyle{}\sum_{k=100}^{200}k. \text{(Use Table 2.)}$
#279
190
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Use the technique given in question $35,$ together with the result of question $37b,$ to derive the formula for $\displaystyle{}\sum_{k=1}^{n}k^{2}$ ... $a_{k} = k^{3}$ in the telescoping sum in question $35.]$
#280
251
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Sum both sides of the identity $k^{2}-(k-1)^{2} = 2k-1$ from $k=1$ to $k=n$ and use question $35$ ... the first $n$ odd natural numbers).a formula for $\displaystyle{}\sum_{k = 1}^{n} k.$